To solve the problem from the image:

We are given:

• $P(A) = 0.64$
• $P(A \text{ and } B) = 0.12$

We need to find $P(B)$. Since $A$ and $B$ are independent events, the following formula applies:

$P(A \text{ and } B) = P(A) \times P(B)$

Now, we can solve for $P(B)$:

$P(B) = \frac{P(A \text{ and } B)}{P(A)} = \frac{0.12}{0.64}$

Let me calculate that for you.

The probability $P(B)$ is $0.1875$.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What does it mean for two events to be independent?
2. How would you calculate $P(A \text{ or } B)$ if events A and B were mutually exclusive?
3. If $P(A) = 0.64$ and $P(B) = 0.1875$, what is $P(A \cup B)$ for independent events?
4. How does conditional probability differ from independent probability?
5. How would this problem change if $A$ and $B$ were dependent events?

Tip: For independent events, the occurrence of one does not affect the probability of the other happening.